Fully isogeometric modeling and analysis of nonlinear 3D beamswith spatially varying geometric and material parameters

Oliver Weegera,∗, Sai-Kit Yeungb, Martin L. Dunnc

aSingapore University of Technology and Design, Information Systems Technology and Design Pillar& SUTD Digital Manufacturing and Design Centre, 8 Somapah Road, Singapore 487372, Singapore

bHong Kong University of Science and Technology, Division of Integrative Systems and Design,Room 6530, 6/F, School of Engineering, Clear Water Bay, Kowloon, Hong Kong

cUniversity of Colorado Denver, College of Engineering and Applied Science,Campus Box 104, Denver, CO 80217-3364, United States

Abstract

We present a fully isogeometric modeling and simulation method for geometrically exact, nonlinear 3Dbeams with spatially varying geometric and material distributions, both along the beam axis and through itscross-section. The approach is based on the modeling of 3D beams using the Cosserat rod theory and the nu-merical discretization using B-Spline and NURBS parameterizations in an isogeometric collocation method.Transversally varying material constitutions are represented using non-homogeneous, functionally gradedbeam cross-section definitions such as laminates and continuously graded cross-sections. Furthermore, tomodel the axial variation of material and geometry, we introduce the parameterization of cross-section prop-erties as spline curves along the beam centerlines. This fully isogeometric modeling and analysis concept,which is based on spline parameterizations of initial beam centerline curves, kinematic unknowns and axi-ally varying material and geometric parameters, has various practical applications enabled by advances inmanufacturing technology, including multi-material 3D printing and advanced manufacturing of compositeswith automated fiber placement. We verify and demonstrate the modeling and simulation approach usingseveral numerical studies and highlight its practical applicability.

Keywords: Isogeometric analysis, 3D beams, Spatially varying materials, Functionally graded beams,Collocation method

1. Introduction

In recent years, many new possibilities for design and manufacturing of slender and light-weight structureshave emerged through the advancement of additive manufacturing technologies [1]. Existing and potentialapplications range from three-dimensionally (3D) printed micro-structures, which can be used to createmeta-materials with high strength-to-weight ratios [2, 3], to multi-functional and composite materials withlocally defined material properties [4–6] (see Fig. 1a), to active, smart and self-assembling materials andstructures, soft robots, and deployable composite space structures, where compliant components need tobe designed with tailored large deformation behavior [7–12] (see Fig. 1b). In particular, multi-method andmulti-material 3D printing now enable the fabrication of freeform structures with arbitrarily varying materialcompositions, thus opening new perspectives for design and application of spatially varying and compositematerials.

Modeling and design of these structures, especially when they are made from soft materials and subjectto large deformations, calls for advanced simulation methods and computer-aided engineering tools that can

∗Corresponding authorEmail addresses: oliver_weeger@sutd.edu.sg (Oliver Weeger), saikit@ust.hk (Sai-Kit Yeung),

martin.dunn@ucdenver.edu (Martin L. Dunn)

Preprint submitted to Elsevier August 3, 2018

(a) Active shape memory polymer structure withaxially varying Young’s modulus distribution [6]

(b) Direct 4D printed structure with transversally varying laminatecross-section with axially varying layer orientation [10]

Figure 1: Examples of 3D beam structures with spatially varying materials and geometries (both axially varying and transver-sally varying) undergoing large deformations and fabricated through additive manufacturing, here multi-material 3D printing

(a) Rod with transversally varying (TV) material distribu-tion, illustrated by color change from red to blue throughthe cross-section

(b) Rod with axially varying (AV) material distribution andradius, illustrated by color change along the centerline from redto blue and decreasing cross-section radius from back to front

Figure 2: Configuration of the Cosserat rod as a framed curve with centerline r(s) and orthonormal frames R(s) =(g1(s),g2(s),g3(s)) defining the cross-section orientations. Colors illustrate the spatial variation of material properties ofa circular cross-section (a) with transversally varying material through the cross-section and(b) axially varying material andradius along the centerline

efficiently incorporate the complex material constituency. Thus, there is a demand for nonlinear modelingand simulation of 3-dimensional beam structures with spatially varying material properties. In the context ofthis work, we distinguish two types of spatial variations: We consider a beam as transversally varying (TV)when its material properties vary through the cross-section of the beam. This type of variation is often calledfunctionally graded in beam literature [13, 14] and can, for instance, be realized through discrete stacking ofmaterials or composites with varying fiber orientations in multiple layers, or continuous grading of materialdistributions (see Fig. 2a for illustration). Furthermore, an axially varying (AV) beam is characterized byvarying parameters of the (homogeneous or transversally varying) cross-section along the centerline of thebeam, which could be continuously changing material properties such as Young’s modulus or geometricparameters such as radius (see Fig. 2b for illustration).

For modeling and simulation of transversally varying, functionally graded 2D beams, several methodsbased on Euler–Bernoulli and Timoshenko beam theories [15–17] have been proposed and a beam finiteelement for functionally graded, i.e., rectangular, layered, beams was introduced including thermoelasticeffects [18]. Many attempts to model and simulate 3D beams undergoing large elastic deformations arebased on the geometrically exact 3D beam theory, usually referred to as Cosserat rod [19, 20], Reissner [21]or Simo [22] beam theory, which is based on the kinematic description of a beam using its centerline positionand the orientation of the rigid cross-section. For modeling of transversally varying, functionally graded andcomposite cross-sections, several extensions and modifications of the theory have been made so far [14]. Ageometrically exact formulation for multi-layer beams was presented in [23], where moments and rotationscan be discontinuous across layers, thus allowing for the overall beam cross-section to deform. In [24]a variational iteration method was proposed for large deflection analysis of planar curved beams madeof functionally graded materials. This work is closely related to [25, 26], where constitutive coefficientsof functionally graded 3D beams with general cross-section shapes and composite layouts were derived

2

and deformation analysis carried out analytically and via finite element methods (FEM). Further worksinclude the study of nonlinear free vibrations of functionally graded carbon nanotube-reinforced compositebeams [27], as well as Timoshenko-like modeling of initially curved and twisted composite beams [28]. Theextension of geometrically exact beams to finite strain 3D material models was presented in [29].

For modeling and analysis of beams with axially varying material and cross-section properties, such astapered, non-prismatic beams, it has been shown that the shear stress distribution depends on all inter-nal forces and moments, which means that it is not exactly reproduced by common Euler–Bernoulli- andTimoshenko-type beam models [30, 31]. To address this issue, for instance, Murın and co-workers havepresented finite element formulations for linear analysis of 3D beams with varying cross-section radii [32],nonlinear 3D truss elements with varying stiffness [33] and a 2D beam finite element for modeling of func-tionally graded beams with spatially varying properties, including electro-thermal-structural multi-physicsanalysis [34]. More recently, a simple Timoshenko-like model for non-prismatic, planar beams was pre-sented [35].

Many different methods have so far been proposed for the numerical discretization of the geometricallyexact 3D model and its extension to functionally graded cross-sections, such as finite element methods [36–38], finite difference methods [39, 40], and isogeometric collocation methods [41, 42], the latter being thestarting point for this work. Isogeometric analysis (IGA) was first proposed in 2005 [43] and has since gainedsignificant popularity in practically all fields of computational mechanics, especially in structural mechanicswhere many new formulations also for beam models have been proposed [44–47]. More recently, isogeometriccollocation methods have been introduced [48, 49] as an efficient alternative to finite element methods [50]and were successfully applied to beam models [51–53], including the Timoshenko-like non-prismatic beammodel [54], as well as other types of problems in structural and continuum mechanics, such as elastostaticsand dynamics [55], Reissner–Mindlin plates [56], Kirchhoff–Love shells [57], and large deformation elasticitywith contact [58].

The basis of isogeometric methods is the parameterization of geometry and discretization of unknownsusing B-Splines, NURBS and other spline-type functions, which are commonly used in computer-aideddesign (CAD). Here, we not only apply an isogeometric discretization, but extend the concept of splineparameterizations to the modeling of axially varying cross-section properties, i.e., material and geometricparameters that change along the centerline of the beam. We demonstrate this fully isogeometric modelingand simulation approach using an isogeometric collocation method for the Cosserat rod model, but the con-cept could be easily extended to other beam models or discretization methods such as finite elements. Beingapplied to the geometrically exact 3D beam model, our approach does not consider the above-mentioned de-pendence of the shear stress distribution on all internal forces and moments for axially varying cross-sectionproperties. Up to the authors’ knowledge, such a formulation has not yet been developed in the contextof nonlinear, geometrically exact 3D beams and according to [30] this dependence can be neglected if thetapering or, in general, the axial variation of parameters, is small. To validate this assumption, we havecarried out comparisons with the planar beam, small deformation applications investigated in [54], wheregood qualitative and quantitative agreement could be observed with deviations of displacements, forces andmoments being not more than 4%. Though this limits the applicability of our fully isogeometric modelingand simulation approach to structures with small axial variation and shear strains, e.g., slender rods, it isstill beyond the capabilities of existing tools for the design of complex structures that can be fabricatedthrough advanced, additive manufacturing technologies.

The outline of this manuscript is as follows: In Section 2 we describe the geometrically exact Cosserat rodmodel and then outline its numerical discretization using the isogeometric collocation method in Sect. 3. Themodeling and implementation of beams with transversally varying materials and functionally graded cross-sections in terms of their constitutive coefficients is presented in Sect. 4 and subsequently the isogeometricparameterization of axially varying cross-section properties is introduced. In Section 5 a new mixed methodis presented, which resolves shear locking and convergence problems for thin rods with transversally varyingcross-sections. The accuracy and convergence behavior of the numerical methods are investigated andconfirmed in Sect. 6, and further applications of the modeling and simulation approach with rods and rodsstructures with TV and AV material and geometric parameters are presented. The paper concludes with asummary of methods and results presented Sect. 7.

3

2. Cosserat rod model

In this Section we briefly introduce the Cosserat rod model, which we use for the mechanical descriptionof slender, elastic, 3-dimensional rods [19, 20, 22]. The Cosserat rod theory can be seen as a nonlinear,geometrically exact extension of the spatial Timoshenko beam model, and is thus also based on the assump-tion that the cross-sections remain undeformed, but not necessarily normal to the tangent of the centerlinecurve, which accounts also for shear deformation.

In the Cosserat model, a rod is represented as a framed curve (see Fig. 2) and thus its configuration iscompletely described by its centerline curve, i.e., the line of its mass centroids r : [0, L]→ R3, and a frame,or local orthonormal basis field R : [0, L] → SO(3). The centerline curve is arc-length parameterized inthe initial configuration given by r0 and R0, which means that ‖r′0(s)‖ = ‖dr0ds ‖ = 1 ∀s ∈ [0, L] and thus

L =∫ L0‖r′0(s)‖ds is the length of the curve. The local frames describe the evolution of the orientation of the

cross-section and can be associated with 3D rotation matrices R(s) = (g1(s),g2(s),g3(s)) ∈ R3×3 : R>R =I, detR = 1 ∀s ∈ [0, L]. As in [42], we use unit quaternions, i.e., normalized quadruples of real numbersq = (q1, q2, q3, q4)> ∈ R4 : ‖q‖ = 1, for the parameterization of frames resp. rotation matrices:

q : [0, L]→ SO(3) ; R(s) = R(q(s)), (1)

where

R(q) =

q21 − q22 − q23 + q24 2(q1q2 − q3q4) 2(q1q3 + q2q4)2(q1q2 + q3q4) −q21 + q22 − q23 + q24 2(q2q3 − q1q4)2(q1q3 − q2q4) 2(q2q3 + q1q4) −q21 − q22 + q23 + q24

(2)

Based on the current (deformed) centerline r(s) and rotation matrix R(s) associated with the frame, aswell as the initial (undeformed) configuration r0(s) and R0(s), the kinematics of the Cosserat rod can bederived. Dropping the dependency on the arc-length parameter s in the notation, the translational strainsare given in the initial, material configuration as:

ε = R>r′ −R>0 r′0. (3)

Using the curvature vector of the rod:

k =

g′>2 g3

g′>3 g1

g′>1 g2

⇔ [k]× = R′>R, (4)

where [·]× represents the skew-symmetric cross-product matrix, the rotational strains are defined as:

κ = k− k0. (5)

With these two nonlinear strain vectors and a commonly used linear elastic constitutive law, whichrestricts the model to large deformations and rotations, but small strains and stresses, the translational androtational stress resultants in the material configuration can be computed as:

σ = A ε + B κ,

χ = B>ε + C κ.(6)

The constitutive matrices A,B,C ∈ R3×3 depend on the cross-section geometry and material, and will bedescribed in detail in Sect. 4 for spatially varying and functionally graded cross-sections. The stress vectorsdefined in (6) are now rotated from the local into the global Euclidian coordinate frame, or in other wordsthey are transformed from the material to the spatial configuration:

n = Rσ,

m = Rχ.(7)

4

0.0 0.25 0.5 0.75 1.0Parameter t 2 [0,1]

0.0

0.25

0.5

0.75

1.0

Bas

is f

unct

ions

Ni(t

)

(a) B-Spline basis functions

centerlinecontrol points & polygoncross-section orientationsimages of collocation pointscross-sections

*

*

(b) Isogeometric parameterization of rod with B-Spline curve

Figure 3: Isogeometric rod parameterization; cubic B-Spline basis functions as shown in (a) with p = 3,m = 11, n = 7,Ξ =0, 0, 0, 0, 1

4, 12, 34, 1, 1, 1, 1 are used for the isogeometric parameterization of the Cosserat rod in (b) with centerline and rotation

quaternions as B-Spline curves (see [42])

Finally, the governing equations of the mechanics of the Cosserat rod model are formulated in terms ofequilibria of linear and angular momentum in the current, spatial configuration:

n′ + n = 0 ∀s ∈ (0, L),

m′ + r′ × n + m = 0 ∀s ∈ (0, L).(8)

Here n and m are external distributed forces and moments. Furthermore, these differential equations haveto be completed with appropriate boundary conditions at the two ends of the rod for s = 0 and s = L.Fixed displacements r and rotations q are specified as Dirichlet boundary conditions r(s) = r, q(s) = q ats = 0, L and forces n and moments m as Neumann boundary conditions n(s) = n, m(s) = m at s = 0, L.Additionally, a unit length constraint for quaternions must hold to complete the equilibrium equations:q>q− 1 = 0 ∀s ∈ [0, L].

3. Isogeometric collocation method

For the numerical discretization of the Cosserat rod model presented above we employ an isogeometriccollocation method. The approach is based on the parameterization of the unknown fields r(s) and q(s)using spline functions and the collocation of the equilibrium equations (8). This method was introducedand investigated in [42], and here we briefly review the basic approach.

3.1. Spline parameterization of Cosserat rods

The basis of any isogeometric method is the parameterization of geometry and unknowns using B-Splinesand Non-Uniform Rational B-Splines (NURBS), which are widely used in computer-aided design [43, 59].Definitions and properties of B-Splines and NURBS can be found in detail in [60]. Here, we briefly introducethe main terminology used associated with splines.

B-Splines are piece-wise polynomial functions and Non-Uniform Rational B-Spline (NURBS) piece-wiserational function of degree p and order p + 1. With Ni(ξ) : Ω0 → [0, 1], i = 1, . . . , n we denote the spline(B-Spline or NURBS) basis functions, which are defined on the parameter domain Ω0 = [ξ1, ξm] ⊂ R usinga knot vector Ξ = ξ1, . . . , ξm with m = n + p + 1, i.e., a non-decreasing sequence of knots ξi ∈ R (i =1, . . . ,m) , ξi ≤ ξi+1 (i = 1, . . . ,m − 1). For two distinct knots ξi 6= ξi+1 the half-open interval [ξi, ξi+1) iscalled the i-th knot span or element and the total number of nonzero knot spans or elements in Ξ is denotedby `. Typically, only open knot vectors are used in IGA, which means that the first and last knot havemultiplicity p+ 1, while inner knots have multiplicity 1 ≤ k ≤ p.

5

The geometry description of a Cosserat rod can now be expressed using spline curves for the initialcenterline r0 and rotation quaternions q0:

r0 : Ω0 → R3, r0(ξ) =

n∑i=1

Ni(ξ) r0,i,

q0 : Ω0 → R4, q0(ξ) =

n∑i=1

Ni(ξ) q0,i, ‖q0(ξ)‖ = 1,

(9)

with control points r0,ii=1,...,n, r0,i ∈ R3 and q0,ii=1,...,n, q0,i ∈ R4.

For illustration, the parameterization of a rod using cubic B-Spline basis functions (p = 3) with n = 7control points and ` = 4 elements in the knot vector Ξ = 0, 0, 0, 0, 14 ,

12 ,

34 , 1, 1, 1, 1 is shown in Fig. 3.

Figure 3a shows the basis functions and Fig. 3b the rod itself in terms of its centerline curve and cross-section frames.

Since the centerline is now parameterized as a spline curve r0(ξ) : Ω0 → R3 with an arbitrary domain ofparameterization Ω0 ⊂ R, it is general not arc-length parameterized. Thus the derivatives of any vector fieldξ → y(ξ) : [0, 1]→ Rd required for evaluation of the Cosserat rod model need to be converted to arc-lengthparameterization using:

y′ =dy

ds=dy

dξ

dξ

ds= y

(ds

dξ

)−1= y

1

‖r0(ξ)‖=

1

Jy, (10)

with y := dy/dξ and J(ξ) := ‖r0(ξ)‖.

3.2. Collocation of strong form of equilibrium equations

As in isoparametric finite elements, the centerline position r and rotation quaternion q in the cur-rent/deformed configuration are now discretized as spline curves rh and qh, just like their initial counterpartsr0 and q0 in (9):

rh : Ω0 → R3, rh(ξ) =

n∑i=1

Ni(ξ) ri,

qh : Ω0 → R4, qh(ξ) =

n∑i=1

Ni(ξ) qi, ‖qh(ξ)‖ = 1.

(11)

The basis functions Ni here refer to either the same or p-/h-/k-refined versions of the ones in (9) and thecontrol points ri ∈ R3 and qi ∈ R4 are arranged in two vectors ~r = (ri)i=1,...,n ∈ R3n and ~q = (qi)i=1,...,n ∈R4n.

Now collocation of the strong form of the equilibrium equations of the Cosserat rod is applied, i.e.,the discretization from (11) is substituted into the balance equations (8) and the quaternion normalizationcondition, which are then evaluated at a set of collocation points τii=1,...,n:

fn(τi) := n′(τi) + n(τi) = 0,

fm(τi) := m′(τi) + r′h(τi)× n(τi) + m(τi) = 0,

fq(τi) := qh(τi)>qh(τi)− 1 = 0.

(12)

At the boundary, i.e., for i = 1 and i = n, the above-mentioned equations are replaced with the evaluationsof the boundary conditions. In order to guarantee the stability of the method, the collocation points arechosen as the Greville abscissae of the spline knot vector [48], which are defined as:

τi =ξi+1 + . . .+ ξi+p

p, i = 1, . . . , n. (13)

6

With (12) we have defined a nonlinear system of 7n equations for the 7n unknowns, i.e., the controlpoint vectors ~r for rh and ~q for qh, which we can write as f : R3n × R4n → R7n,

f(~r, ~q) =

fn(τi)fm(τi)fq(τi)

i=1,...,n

(~r, ~q) = 0. (14)

This nonlinear system is then solved with a Newton’s method, which requires also the evaluation of thetangent stiffness matrix K(~r, ~q) = df/d(~r,~q) for linearization, see [42] for details.

In [42] it is furthermore shown how the isogeometric collocation formulation can be extended to rodstructures, where several rods can be interconnected. The rods are coupled rigidly, which means that changeof position and rotation have to be equal for all rods connected at an interface and that the equilibria oflinear and angular momentum must hold at the interface.

4. Isogeometric modeling of rods with spatially varying geometric and material parameters

Typically, the cross-sections of Cosserat rods as introduced above, or in general also in other 2D and3D beam models, are assumed to be made from a single material and to be constant over the length of thebeam. However, the constitutive relationship established in (6) is not restricted to such cases and can alsoadmits transversally (TV) and axially varying (AV) rod cross-sections, where the material composition canchange throughout the cross-section and material and geometry can both vary along the centerline. In thefollowing, we address those more general cases through the isogeometric analysis framework using splineparameterizations.

4.1. Constitutive coefficients for transverse variations

The constitutive equation (6) defines a linear relationship between strain and stress resultant of the rodmodel, and reads in its most general formulation as:(

σχ

)=

(A B

B> C

)(εκ

), (15)

where the constitutive matrices A, B, C ∈ R3×3 take the following form:

A =

A11 A12 0A12 A22 00 0 A33

, B =

0 0 B13

0 0 B23

B31 B32 0

, C =

C11 C12 0C12 C22 00 0 C33

. (16)

Their coefficients depend on the geometric layout and material constitution of the beam cross-section. Adetailed derivation of the coefficients can be found in [25] and takes into account that the Bernoulli hypothesismust be fulfilled, i.e., that plain cross-sections must remain plain under deformation.

In the following, we restrict ourselves to cases where A12 = B13 = B23 = 0 and according to [25] theremaining coefficients can be computed as:

A11 = k1

∫S

G(x) dx, A22 = k2

∫S

G(x) dx, A33 =

∫S

E(x) dx,

B31 =

∫S

E(x)x2 dx, B32 =

∫S

E(x)x1 dx, C11 =

∫S

E(x)x22 dx,

C22 =

∫S

E(x)x21 dx, C12 =

∫S

E(x)x1x2 dx, C33 ≡ C33(S,E).

(17)

Here, k1 and k2 are the shear correction factors, S is the geometric domain of the cross-section, and G =E/(2 + 2ν) and E are the shear and Young’s modulus and ν is the Poisson’s ratio, which all depend on the

7

(a) rectangular bilayer (b) circular bilayer (c) continuously graded (d) rotated

Figure 4: Definition and geometric parameters of transversally varying (functionally graded) cross-section types: (a) rectangularand (b) circular bilayer cross-sections, (c) rectangular continuously graded cross-section (here linearly graded), and (d) rotatedcross-section (here circular bilayer)

material point x ∈ S inside the cross-section domain S, which is aligned such that x1 denotes the coordinatealong the g1-direction and x2 along the g2-direction.

For the conventional case of homogeneous, isotropic cross-sections, where E and G are constant through-out the cross-section, this simplifies to:

A =

k1GA 0 00 k2GA 00 0 EA

, B ≡ 0 , C =

EI1 0 00 EI2 00 0 GJ

, (18)

where A is the area, I1 and I2 the second moments of area, and J the polar moment of area of S.

4.1.1. Bilayer laminate cross-sections

As a first type of laminate or composite cross-section, we present the coefficients for bilayer cross-sectionswith either rectangular or circular shape, see Fig. 4a-b.

For the following derivations, a few basic definitions and assumptions are made: the geometric dimensionsof the cross-section S ⊂ R2 are width b and height h for rectangular and radius r for circular shape; the layersare separated along the local g2-direction; the layering is determined by the layer height hL ∈ (−h/2, h/2)or hL ∈ (−r, r), respectively; the domain of the lower layer is denoted as S0 = x ∈ S : x2 ≤ hL withYoung’s modulus E0; the domain of the upper layer is denoted as S1 = x ∈ S : x2 > hL with Young’smodulus E1; for both layers Poisson’s ratios ν = ν0 = ν1 and mass densities ρ = ρ0 = ρ1 are equal.

Then, the constitutive matrices can be written as:

A =

k1µ 0 00 k2µ 00 0 1

A33, B =

0 0 00 0 0B31 0 0

, C =

C11 0 00 C22 00 0 C33

(19)

with k1 = k2 = 56 , µ = 1

2(1+ν) , and:

A33 = E0 A033 + E1 A

133, A0

33 =

∫S0

dx, A133 =

∫S1

dx,

B31 = E0 B031 + E1 B

131, B0

31 =

∫S0

x2 dx, B131 =

∫S1

x2 dx,

C11 = E0 C011 + E1 C

111, C0

11 =

∫S0

x22 dx, C111 =

∫S1

x22 dx,

C22 = E0 C022 + E1 C

122, C0

22 =

∫S0

x21 dx, C122 =

∫S1

x21 dx,

C33 ≡ C33(S0, E0, S1, E1).

(20)

8

The detailed calculation of those coefficients and integrals for both cases of rectangular and circular bilayercross-sections, please refer to the Appendix A.1.

4.1.2. Continuously varying cross-sections

Next, we introduce the commonly addressed case of a rectangular, continuously functionally gradedcross-section, see Fig. 4c.

Here, the Young’s modulus varies throughout the cross-section as a function of the x2-coordinate:

E(x2) = E0 + (E1 − E0)

(1

2+x2h

)p, (21)

where x2 ∈ [−h/2, h/2] and p > 0 is the exponent of a power law. This yields an asymmetric Young’s modulusdistribution with respect to the x1-axis of the cross-section and thus to the same general representationof the constitutive matrices as in (19) and (20). The detailed coefficients are derived and calculated inAppendix A.2.

4.1.3. Rotated cross-sections

To further generalize the transverse variation of cross-section types we now introduce rotated cross-sections, where for instance the orientation of the layering or gradient described in the preceding subsectionsis rotated by an angle α ∈ R around the local g3-axis, see Fig. 4d.

For any point x = (x1, x2)> ∈ S, where S is the rotated version of an axis-aligned cross-section S0 withlayered or graded Young’s modulus distribution E(x0), this means that:

x = ϕ(x0) = Rα x0 =

(cosα − sinαsinα cosα

)x0, (22)

where ϕ : S0 → S is an injective and differentiable mapping. Then we can compute the constitutive coeffi-cients of the rotated cross-section S = ϕ(S0) using integration by substitution with |detDϕ| = |detRα| = 1:

A33 =

∫S

E(x) dx =

∫S0

E(x0) dx0 = A033,

B31 =

∫S

E(x)x2 dx =

∫S0

E(x0)(x01 sinα+ x02 cosα) dx0 = cosα B031 − sinα B0

32,

B32 =−∫S

E(x)x1 dx =−∫S0

E(x0)(x01 cosα− x02 sinα)dx0 = sinα B031 + cosα B0

32,

C11 =

∫S

E(x)x22 dx =

∫S0

E(x0)(x01 sinα+ x02 cosα)2 dx0 = cos2 α C011 − 2 cosα sinα C0

12 + sin2 α C022,

C22 =

∫S

E(x)x21 dx =

∫S0

E(x0)(x01 cosα− x02 sinα)2 dx0 = sin2 α C011 + 2 cosα sinα C0

12 + cos2 α C022,

C12 =−∫S

E(x)x1x2 dx =−∫S0

E(x0)(x01 cosα−x02 sinα)(x01 sinα+x02 cosα)dx0

= cosα sinα (C011 − C0

22) + (cos2 α− sin2 α)C012,

C33 = C033.

(23)More compactly, we can write in matrix form:

A = A0 (= RαA0R>α ), B = B0R>α (= RαB

0R>α ), C = RαC0R>α , (24)

9

slightly abusing the notation since Rα is now the extended 3-dimensional rotation matrix:

Rα =

cosα − sinα 0sinα cosα 0

0 0 1

. (25)

4.2. Isogeometric parameterization of axially varying cross-section parameters

As defined earlier, an axially varying beam is characterized by varying properties of the cross-sectionalong the centerline of the beam, see Fig. 2b. In general, we can consider both material and geometricparameters as axially varying. This means that properties such as Young’s modulus E, Poisson’s ratio ν,radius r of a circular cross-section, layer height hL of a bilayer cross-section, or even the general cross-sectionshape S can be expressed as a function of the arc-length parameter s of the rod.

Summarizing all those cross-section parameters in one vector u = (E, ν, r, . . .)> ∈ Rdu , we can parame-terize the axial variation of the cross-section properties as a NURBS curve:

u(s) =

nu∑i=1

Nui (s) ui. (26)

In this isogeometric parameterization, du is the number of design parameters, Nui are nu NURBS basis

functions with knot vector Ξu of degree pu with `u elements, and ui are the nu control points of the curve.This NURBS basis can be chosen arbitrarily, depending on the design of the axially varying rod, i.e., it doesnot necessarily have to be the same as the one used to parameterize and discretize the rods in (9) or (11).

For the constitutive matrices and coefficients, as introduced above in Sect. 4.1 for both homogeneous andtransversally varying cross-sections, the parameterization of the cross-section properties implies, of course,that they also become dependent on s:

A(s) ≡ A(u(s)), B(s) ≡ B(u(s)), C(s) ≡ C(u(s)). (27)

The axially varying constitutive matrices are used in the constitutive law (6) to compute the stress resultantsfrom the strains, which are then transformed to the material configuration in (7) and finally used in theequilibrium equations (8). Since the balance equations require the arc-length derivation of the internal forcesand moments:

n′ = (Rσ)′ = R′σ + Rσ′,

m′ = (Rχ)′ = R′χ+ Rχ′,(28)

it follows for an axially varying rod that also the constitutive matrices have to be differentiated, since theyare not constant anymore:

σ′ = (Aε+ Bκ)′ = A′ε+ Aε′ + B′κ+ Bκ′,

χ′ = (B>ε+ Cκ)′ = B′>ε+ B>ε′ + C′κ+ Cκ′.(29)

Since the constitutive matrices are computed from the vector u(s), their derivatives A′, B′ and C′ can beeasily evaluated through the isogeometric parameterization, e.g.:

A′ =dA

ds=

du∑i=1

dA

duiu′i. (30)

5. Mixed method for slender composite rods

Shear locking is a well-known phenomenon that occurs when shear compliant beam theories are employedfor slender beams, i.e., beams with small thickness-to-length ratios (r/L 1). It manifests itself through

10

instability of the numerical method, loss of accuracy and deterioration of convergence rates. Here, we addressthis issue in the context of rods with transversally varying, functionally graded cross-sections.

A mixed method that cures shear locking for isogeometric collocation of the Timoshenko beam modelwas already introduced in [51] and theoretically investigated in [53]. The approach was also extended to theCosserat rod model in [42, 61] and is based on an independent discretization of the internal forces n andmoments m:

nh : Ω0 → R3 : nh(ξ) =

n∑i=1

Ni(ξ)ni, mh : Ω0 → R3 : mh(ξ) =

n∑i=1

Ni(ξ)mi, (31)

thus introducing additional unknowns with the force and moment control points ni and mi (i = 1, . . . , n),and consequently also requiring modified and additional collocated balance equations:

fn(τi) := n′h(τi) + n(τi) = 0,

fm(τi) := m′h(τi) + r′h(τi)× nh(τi) + m(τi) = 0,

fq(τi) := qh(τi)>qh(τi)− 1 = 0,

fu(τi) := nh(τi)− (Rσ)(τi) = 0,

fv(τi) := mh(τi)− (Rχ)(τi) = 0.

(32)

Compared to the primal formulation using only displacement and rotation degrees of freedom in (12), herethe independent forces nh and moments mh are used in the equations fn and fm and in the additionalequations fu and fv they are set equal to their counterparts n = Rσ and m = Rχ computed from theprimary variables.

This approach cures the shear locking problem for beams with homogeneous cross-sections, since theill-conditioning of stress components σ = Aε and χ = Cκ, which is caused by the difference in scalesbetween the constitutive matrices A ∼ r2 and C ∼ r4, is resolved by the independent discretization offorces nh = Rσ and moments mh = Rχ. However, when transversally varying, non-symmetric cross-sections are used, the constitutive equations additionally include the translational–rotational coupling termswith the matrix B, which scales as B ∼ r3, see (A.2). This introduces an additional imbalance of the scaleswithin the constitutive equations for both, the translational and rotational stresses. Thus, shear lockingand numerical instability can still be present when the above-mentioned mixed method is used for slender,functionally graded rods, see numerical examples in Sect. 6.1.

To cure shear locking also for these cases, we introduce an enhanced mixed method, in which thetranslational and rotational strain contributions to the internal stresses are separated:

σε := A ε, σκ := Bκ, χε := B>ε, χκ := Cκ. (33)

Then, the corresponding internal force and moment contributions are discretized independently:

nεh : Ω0 → R3 : nεh(ξ) =

n∑i=1

Ni(ξ)nεi , nκh : Ω0 → R3 : nκh(ξ) =

n∑i=1

Ni(ξ)nκi ,

mεh : Ω0 → R3 : mε

h(ξ) =

n∑i=1

Ni(ξ)mεi , mκ

h : Ω0 → R3 : mκh(ξ) =

n∑i=1

Ni(ξ)mκi .

(34)

In addition to the 7n primal unknowns, i.e., the control points for rh and qh, we know have 4 · 3n unknownsfor the control points nεi ,nκi ,mε

i ,mκi i=1,...,n and the corresponding collocated balance equations become:

11

fn(τi) :=(nεh(τi) + nκh(τi)

)′+ n(τi) = 0,

fm(τi) :=(mεh(τi) + mκ

h(τi))′

+ r′h(τi)×(nεh(τi) + nκh(τi)

)+ m(τi) = 0,

fq(τi) := qh(τi)>qh(τi)− 1 = 0,

f εu (τi) := nεh(τi)− (Rσε)(τi) = 0,

f κu (τi) := nκh(τi)− (Rχκ)(τi) = 0,

f εv (τi) := mεh(τi)− (Rσε)(τi) = 0,

f κv (τi) := mκh(τi)− (Rχκ)(τi) = 0.

(35)

Altogether, (35) forms a nonlinear system of 19n equations and unknowns:

f(~r, ~q, ~nε, ~nκ, ~mε, ~mκ) =((

fn, fm, fq, fεu , f

κu , f

εv , f

κv

)(τi))>i=1,...,n

= 0. (36)

This nonlinear system is again solved with a Newton’s method.To achieve optimal convergence rates, the same basis functions Ni are used for the discretization of

nεh, nκh, m

εh and mκ

h as were used for rh and qh and all equations are collocated at the original collocationpoints τi [51]. Furthermore, much like the initial mixed method shown in (31), the continuity requirementsfor the basis functions are reduced, since both, the primal and secondary unknowns, only have to be differ-entiated once instead of twice, as in the original method (12). This also reduces the computational effort forthe assembly of the system and its Jacobian matrix, since second derivatives do not have to be evaluatedanymore. The split of translational and rotational parts of internal forces and moments in the enhancedmixed method introduces additional computational effort, since more discretized variables need to be eval-uated (e.g., nεh and nκh instead of nh), transformed (e.g., Rσε and Rσκ instead of Rσ), and assembled intoseparate positions in the force vector and stiffness matrix. Also, the computational effort for solving thelinear systems within the Newton’s method increases, of course, since the system size increases from 7n forthe primal method and 13n for the simpler mixed method to 19n for the enhanced mixed method.

However, the main objective and advantage of the mixed methods, and particularly the novel enhancedmixed method for transversally varying cross-sections introduced here, is that they cure shear locking andthus provide higher accuracy and increased stability compared to the primal or simple mixed method. Thismeans that a smaller degree or number of elements can be used to obtain same accuracy and less load stepsand less iterations per load step are required to obtain convergence, which makes the mixed methods overallcomputationally much more efficient, as will be shown in Sect. 6.1.2.

6. Numerical results

In this Section we apply isogeometric parameterizations of axially and transversally varying cross-sectionstogether with the isogeometric collocation methods for nonlinear 3D composite rods in several computa-tional examples. We focus on both, the validation of the approach using convergence studies and referenceexamples, as well as advanced applications that highlight the capabilities and efficiency of the isogeometricmodeling and analysis concept.

6.1. Convergence studies

First, we investigate the effect of employing axially varying parameters and transversally varying, func-tionally graded cross-sections on the general accuracy and convergence behavior of the isogeometric collo-cation methods presented.

6.1.1. Axially varying cross-section parameters

As reference set-up, we take a rod of length L = 1.0 m with a circular cross-section of radius R = 0.05m. The initial, constant material parameters are Young’s modulus E(s) ≡ E0 = 100 MPa and Poisson’s

12

Figure 5: Rods with axially varying material parameters. Initial rods with axially varying Young’s modulus distributions ontop (colored by Young’s modulus) and deformed configurations below (colored by curvature strain)

ratio ν = 0.45, which we refer to as case (a). The rod is clamped at one end and a moment m(L) = πE0I isapplied at the other over 8 load steps, deforming it into a semicircle, see Fig. 5a. Now, we introduce threevariations of the above-mentioned rod with axially varying Young’s modulus, which are parameterized asfollows using u(s) ≡ E(s), see also Fig. 5b-d:

(b) p = 3, ` = 1, n = 4, ~u = E0 · (1.0, 0.0, 2.0, 1.0). The Young’s modulus is parameterized globally alongthe rod as a polynomial of degree 3, since the B-Spline has only 1 element.

(c) p = 3, ` = 4, n = 7, ~u = E0 · (1.0, 1.0, 0.4, 1.4, 1.2, 0.7, 1.0). The Young’s modulus is parameterizedarbitrarily along the rod as a B-Spline of degree 3 with 4 elements (knot spans).

(d) The Young’s modulus is parameterized locally, using the same B-Spline basis employed for the spatialdiscretization of the rod and ~u is chosen such that E(s) interpolates a sinusoidal curve, i.e., E(τi) =E0 · (1− 0.3 sin 2πτi).

For all four cases of Young’s modulus parameterizations, we apply the same moment to deform themand compute a reference solution r∗ with a highly refined isogeometric discretization using p = 10, ` = 128.Then, we evaluate the L2-errors of deformed centerline positions eh = ‖rh − r∗‖, where rh indicates thesolutions with lower levels of k-refinement obtained for p = 3, . . . , 8 and ` = 4, . . . , 64. The correspondingconvergence plots for eh over number of elements (knot spans) ` are shown in Fig. 6.

Figure 6a refers to the conventional case of a rod with constant cross-section parameters, i.e., hereconstant Young’s modulus. The convergence rates that can be observed here correspond to the typical onesto be expected for an isogeometric collocation method, as already presented for Cosserat rods in [42].

For the global parameterization with degree 3, which is shown in Fig. 6b, the convergence rates are wellreproduced, but the error constants are about 1–3 orders of magnitude higher due to the variation of theYoung’s modulus. This shows that the collocation method can deal well with those global cross-sectionparameterizations and there is no negative effect on convergence rates. Such behavior can be expectedfor a Galerkin method, since the element-wise integration can also exactly integrate the axially varyingmaterial parameters (if the degree of the spatial discretization is not less than the one of the materialparameterization). However, in the case of a collocation method, where there is no fixed distribution ofcollocation points over elements, such behavior is not immediately clear from the theoretical point of view.

When the arbitrary parameterization over 4 elements with degree 3 is applied, see Fig. 6c, a negativeeffect on both error constants as well as convergence rates of higher order discretizations with p ≥ 6 canbe observed, since the collocation points of the spatial discretization are not compatible with the Young’smodulus parameterization. Furthermore, the absolute errors are again higher, since the axially varyingparameterization of the cross-section is “rougher” due to the choice of control points in ~u.

The Young’s modulus distribution in case (d) is actually very similar to case (b), but interpolating thesinusoidal shape of E(s) with the same basis functions as used in the spatial discretizations. As shown inthe convergence plots in Fig. 6d, the absolute errors are also similar to (b), however, especially for higherorder and higher number of elements it seems that the theoretical convergence rates can be more closely

13

22 23 24 25 2610−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

number of elements (knot spans) `

L2-e

rror

ofce

nte

rlin

e

p = 3p = 4p = 5p = 6p = 7p = 8

C2 `−2

C4 `−4

C6 `−6

C8 `−8

(a) E(s) = E0 is constant

22 23 24 25 2610−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

number of elements (knot spans) `

L2-e

rror

ofce

nte

rlin

e

p = 3p = 4p = 5p = 6p = 7p = 8

C2 `−2

C4 `−4

C6 `−6

C8 `−8

(b) E(s) of degree 3 with 1 element

22 23 24 25 2610−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

number of elements (knot spans) `

L2-e

rror

ofce

nte

rlin

e

p = 3p = 4p = 5p = 6p = 7p = 8

C2 `−2

C4 `−4

C6 `−6

C8 `−8

(c) E(s) of degree 3 with 4 elements

22 23 24 25 2610−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

number of elements (knot spans) `

L2-e

rror

ofce

nte

rlin

e

p = 3p = 4p = 5p = 6p = 7p = 8

C2 `−2

C4 `−4

C6 `−6

C8 `−8

(d) E(s) interpolating sinusoidal curve

Figure 6: Convergence study for axially varying Young’s modulus distribution. Convergence plots show L2-error in deformedcenterline position eh = ‖rh − r∗‖ for all four cases (a)–(d) of Young’s modulus parameterizations

obtained. This behavior is reasonable, since the local material parameterization is collocated at suitablepoints, but there is an additional error component stemming from the interpolation of the sinusoidal curve.

Altogether, we can conclude from these convergence studies that the isogeometric collocation methodstill provides an accurate and efficient discretization method when AV cross-section parameters are used.Most importantly, the most convincing feature of collocation is preserved, i.e., high accuracy can be obtainedfor a low number of degrees of freedom when a higher order discretization with a relative small number ofelements is employed.

Remark 1. Here, we have used a beam with a fairly large thickness-to-length ratio, i.e., 2r/L = 0.1, sincewe wanted to avoid that shear locking affects the results. As mentioned above in Sect. 5, mixed methods canbe used to eradicate the effects of shear locking and improve convergence rates for odd spline basis functiondegrees and general accuracy, i.e., error constants.

6.1.2. Slender rods with transversally varying materials

The use of functionally graded cross-sections with transversally varying materials does in general notaffect the accuracy and convergence behavior of the discretization method, since the constitution of thecross-section only defines the constitutive parameters, see Sect. 4.1. However, when the Cosserat rod is

14

22 23 24 25 2610−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

number of elements (knot spans) `

L2-e

rror

ofce

nte

rlin

e

(a1) R = 0.05 m, primal

22 23 24 25 26

number of elements (knot spans) `

(a2) R = 0.05 m, mixed

22 23 24 25 26

number of elements (knot spans) `

p = 3p = 4p = 5p = 6p = 7p = 8

C2 `−2

C4 `−4

C6 `−6

C8 `−8

(a3) R = 0.05 m, enhanced mixed

22 23 24 25 2610−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

number of elements (knot spans) `

L2-e

rror

ofce

nte

rlin

e

(b1) R = 0.005 m, primal

22 23 24 25 26

number of elements (knot spans) `

(b2) R = 0.005 m, mixed

22 23 24 25 26

number of elements (knot spans) `

p = 3p = 4p = 5p = 6p = 7p = 8

C2 `−2

C4 `−4

C6 `−6

C8 `−8

(b3) R = 0.005 m, enhanced mixed

Figure 7: Convergence study for slender rods with transversally varying materials. Convergence plots show error in centerlineposition for (a) R = 0.05 m and (b) R = 0.005 m using (1) primal (r,q)h, (2) mixed (r,q,n,m)h, and (3) enhanced mixed(r,q,nε,nκ,mε,mκ)h methods

slender, a mixed method should or has to be used to resolve the shear locking problem. Having introducedan extension of the mixed collocation method in Sect. 5, we now evaluate its performance for the analysisof slender rods with transversally varying material compositions.

Again, we take a rod of length L = 1.0 m, which now has a circular, bilayer laminate cross-section, seeSect. 4.1.1. The material parameters are chosen as Young’s moduli E0 = 100 MPa, E1 = 10 MPa andPoisson’s ratios ν0 = ν1 = 0.45. Now, the radius of the cross-section is varied using (a) R = 0.05 m and (b)R = 0.005 m, while the relative layer ratios are kept constant using layer heights hL = −0.5R. Like before,the rods are clamped at one end and moments m(L) = 0.5πC11 (which depend on R, see (20)) are appliedat the other end incrementally over 8 load steps, deforming each rod to almost a semicircle.

For each choice of R, we use the conventional formulation, where only centerline positions and quaternionsare discretized (primal (r,q)h, see Sect. 3.2), the original mixed method from [42], where additionallythe translational and rotational stresses are discretized (mixed (r,q,n,m)h, see Sect. 5, Eqn. (32)), andthe newly presented enhanced mixed method, where the translational and rotational contributions to thetranslational and rotational stresses are separately discretized (enhanced mixed (r,q,nε,nκ,mε,mκ)h, seeSect. 5, Eqn. (35)) to compute the displaced configurations. We evaluate the L2-errors of deformed centerlinepositions against reference solutions with highly refined isogeometric discretizations (p = 10, ` = 128) and

15

per iteration [ms] r = 0.05 [s] r = 0.005 [s]PM MM EM PM MM EM PM MM EM

# load steps - - - 8 18 8 8 129 8p = 4 # iterations - - - 64 246 46 106 2014 48` = 16 assembly 10.2 4.2 7.0 0.680 1.037 0.327 1.029 8.352 0.330n = 20 linear solve 1.5 2.4 4.8 0.099 0.622 0.230 0.155 4.770 0.225

total 11.7 6.6 11.8 0.779 1.659 0.557 1.183 13.121 0.555

# load steps - - - 8 17 8 8 128 8p = 4 # iterations - - - 64 214 47 200 1774 48` = 64 assembly 37.9 24.2 47.2 2.498 5.275 2.261 7.353 42.308 2.222n = 68 linear solve 4.2 6.2 11.6 0.266 1.327 0.542 0.851 11.017 0.557

total 42.1 30.5 58.8 2.764 6.602 2.803 8.204 53.325 2.778

Table 1: CPU times for primal (PM), mixed (MM) and enhanced mixed (EM) methods for slender rods with transversallyvarying materials. Assembly, linear solution and total time are shown for one iteration (in milliseconds), as well as for solvingthe beam displacement in (at least) 8 load steps for r = 0.05 and r = 0.005 (in seconds)

resulting convergence plots are shown in Fig. 7.When the rod becomes more slender, the primal method loses accuracy due to shear locking, see Fig. 7a1

and Fig. 7b1. Furthermore, the convergence behavior of the Newton iteration is affected and a lot ofiterations per load step are required (around 10–20), or the 8 load steps had to be further subdivided toachieve convergence for R = 0.005 m.

The original mixed method, which is shown in Fig. 7a2 and Fig. 7b2, seems to attain high accuracyand good convergence rates for both cross-section radii. However, to obtain these results, we had to furthersubdivide the application of the moment from 8 to 32 and 160 (!) load steps for R = 0.05 m and R = 0.005 m,respectively. Otherwise, the method is numerically unstable and the Newton iterations would not convergeat all. Thus, employing this method for TV cross-sections means a tremendous increase in computationalcost and is not advisable.

For the newly proposed enhanced mixed method, it can be seen in Fig. 7a3 and Fig. 7b3 that almostthe same high accuracy and ideal convergence rates can be achieved independently of the slenderness of thecross-section. Furthermore, the method only needed very few Newton iterations for each of the 8 load steps,typically only 6 iterations for a relative accuracy of 10−9.

In Table 1 we have summarized the CPU times required for assembly and solution for the three methodswith p = 4 and ` = 16, 64, using an Intel c© CoreTM i7-5960X 3.00 GHz PC with 32 GB RAM in single-coremode and UMFPACK direct sparse linear solver [62]. Per iteration, the mixed method (MM) is fasterthan the primal (PM) and enhanced mixed method (EM), but while PM and MM suffer from locking andinstability, and thus need a lot of load steps and iterations, EM is very efficient and overall much faster thanboth PM and MM — and at the same time also more accurate than PM.

Thus, we can conclude that the newly developed enhanced mixed method overcomes the shear lockingproblem and provides a stable and efficient solution process for slender rods with TV cross-sections.

6.2. Applications of rods with spatially varying materials

Next, we study a few simple applications of modeling of 3D beams with spatially varying materials, i.e.,beams with both axially and transversally varying cross-section parameters, including rotated cross-sectionorientations.

6.2.1. Homogeneous vs. graded vs. bilayer cross-sections

Functionally graded, TV cross-sections with asymmetric Young’s modulus distributions, such as thebilayer laminates and continuously graded variations introduced in Sect. 4.1, are characterized by the cou-pling of tension and bending deformations. This enables the design of rods with very specific and tailoreddeformation behavior.

16

Figure 8: Homogeneous vs. graded vs. bilayer cross-sections. Initial rods with (a) homogeneous cross-section, (b) linear trans-verse grading and (c) bilayer laminate (on top, colored by Young’s modulus distribution) and deformed, curved configurationsafter moment was applied (below, colored by curvature strains)

To demonstrate this, we compare the bending behavior of three slender cantilever beams of length L = 1m with rectangular cross-sections with b = 3 mm, h = 2 mm, E0 = 100 MPa, E1 = 10 MPa, and ν = 0.45with (a) homogeneous cross-section with E = 0.5(E0 + E1), (b) continuous, linear grading with p = 1, and(c) bilayer laminate with hL = 0 mm. Consequently, all respective coefficients of the constitutive matrices

A and C are equal for the three cases, but B(a)31 = 0 and B

(c)31 = 3

2B(b)31 .

We apply a moment m(L) = 0.5πC11 at the free end of the beams and discretize them using theenhanced mixed method with p = 6, ` = 16 to compute the deformations. The initial beams and theirdeformed configurations for all cases are shown in Fig. 8. Application of a moment results in constantextensional strains and bending curvatures, which are:

ε(a)3 = 0, ε

(b)3 = 0.0055, ε

(c)3 = 0.0129,

κ(a)1 = 1.571, κ

(b)1 = 2.022, κ

(c)1 = 3.155.

While the homogeneous beam bends by exactly 90, the linearly graded beam bends about 115, and thebilayer beam bends even slightly more than 180. This provides an interesting insight into the behavior andresulting design opportunities of beams with TV material compositions.

Remark 2. In this simple case of 2-dimensional beam bending, the above-computed quantities can also bedetermined analytically through the relationship:(

A33 B31

B31 C11

)(ε3κ1

)=

(0m

).

6.2.2. Bilayer cross-section with axially varying geometric parameters and rotation

Now, we introduce an application where a TV cross-section, here a bilayer laminate, has AV geometricproperties, here cross-section radius, layer height and rotation, see Fig. 9.

The bilayer cantilever beam has length L = 1 m, radius parameter r0 = 0.025 m, Young’s moduliE0 = 100 MPa and E1 = 10 MPa, and ν = 0.45. The following cross-section properties are axiallyvarying, here parameterized linearly from s = 0 to s = 1: radius r(s) = r0(2 − s), layer height ratiohR(s) = 0.25 + 0.5s, and rotation angle α(s) = π s. Thus, we have pu = 1, `u = 1, nu = 2 and theparameter vector is u ≡ (r, hR, α)>. A follower moment m(L) = R>(0.5π C22(L/2), 0, 0)> is applied on itsfree end.

The initial shape and material constitution of the beam and its deformed configuration are visualizedin Fig. 9a-b. As can be seen from the deformed shapes and the values of curvature strains κ1 and κ2, thebeam undergoes a highly complex and nonlinear bending behavior with coupling of both curvature modes,twist and tension. Our isogeometric modeling and analysis framework enables convenient modeling of suchcomplex beams and their accurate analysis, here using the newly introduced enhanced mixed method, ascan be seen from the convergence plot for the L2-error of displacements shown in Fig. 9c.

17

22 23 24 25 2610−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

number of elements (knot spans) `

L2-e

rror

of

cente

rlin

e

p = 3p = 4p = 5p = 6p = 7p = 8

C4 `−4

C6 `−6

C8 `−8

Figure 9: Bilayer cross-section with axially varying geometric parameters and rotation. Initial (colored by Young’s modulus)and deformed (colored by curvature strains) rod configurations in (a) x/z-view and (b) y/-x-view. (c) Convergence of L2-errorof displacements.

6.3. Application to a graded 3D lattice structure

Finally, we apply the concept of AV and TV material compositions to a graded, lattice-like rod structure.This application points towards the design of structures with spatially varying materials for specific defor-mation behavior, instabilities, or energy absorption capabilities, such as the soft, compliant lattice structurespresented in [63].

Here, we introduce a tire-like, cylindrical 3D lattice structure with outer radius 100 mm, inner radius 70mm, and height 30 mm, see Fig. 10. We use a honeycomb-type lattice unit cell structure with 16 cells incircumferential direction of the cylinder. The individual struts have an axially varying radius, which variesin radial direction of the cylindrical geometry from 1.5 mm at the inner side to 1.0 mm at the outer side.Struts at the inner and outer sides of the cylindrical geometry are transversally varying bilayers with heightratio hR = 0.8, where the smaller layer has Young’s modulus E1 = 1.2 GPa and the larger layer and alluniform rods have Young’s modulus E0 = 0.6 MPa. Thus, the small layers act like a very stiff coating onthe otherwise rather soft structure. The Poisson’s ratios are ν0 = ν1 = 0.45. These material parameterscorrespond to typical polymer inkjet 3D printing materials, as for instance used in [63].

For axially varying cross-section properties, here the radius, we simply use a linear global parameteriza-tion, i.e., u ≡ r and pu = 1, `u = 1. The isogeometric discretization of the totally 512 rods is done using themixed methods with p = 6, ` = 8. The original mixed method is used for interior rods with homogeneouscross-sections and the enhanced mixed method is used for the bilayers, since there is a strong difference ofYoung’s moduli of more than three orders of magnitude.

For the mechanical simulation of the structure, the rods at the right end are clamped and a displacementof 40 mm in x-direction is incrementally applied to the rods on the left end over 30 load steps. This resultsin an overall compression of 20% of the structure in x-direction and to complex deformation patterns of theindividual rods, including torsion and double-curved bending deformations, see Fig. 10.

7. Summary and conclusions

We have introduced a fully isogeometric modeling and analysis method for 3D beams with spatiallyvarying material and geometric parameters. The beams were modeled using the geometrically exact, non-linear Cosserat rod theory and discretized by an isogeometric collocation method. Axially varying materialand geometric properties of the beam cross-sections are also parameterized using spline functions, allow-ing a convenient and flexible implementation of the method in the IGA context. Effects of the axiallyvarying parameterizations on convergence behavior were investigated, showing that there are no significantimplications on convergence rates of the collocation method. Transversally varying, functionally graded

18

Figure 10: Graded 3D lattice structure. The initial, tire-like lattice structure is shown on the left side, colored by the Young’smodulus distribution. The deformed structure after application of 40 mm deformation is shown on the right side, colored bythe torsion strain κ3

cross-sections were also implemented as bilayer laminates and with continuous grading of Young’s modulus.To overcome shear locking and convergence issues for those transversally varying beams, a new enhancedmixed collocation method was introduced and successfully numerically evaluated.

The next step based on this work is the implementation of isogeometric design optimization methods, thatoptimize the spatially varying material and geometric parameters of the cross-sections. Furthermore, shapeand (ground-structure) topology optimization could be used to implement an even more flexible optimizationframework for 3-dimensional beam structures. Potential applications for these methods, which would alreadybe realizable with current advanced manufacturing technologies, include design and optimization of soft,multi-material and composite rod structures that are subject to large deformations and instabilities, suchas soft robotic actuators, artificial muscles and flexible space structures.

To overcome the limitations of the Cosserat rod model in terms of the approximation of shear stressdistribution for axially-varying beams, a model could be derived that enables the coupling of all forces andmoments with stress and strain components. This model could then be validated against the current model,a beam model where a continuum hyperelastic constitutive law is used and integrated over the cross-section,see [29], and full 3D finite element analysis.

Acknowledgements

The authors acknowledge support from the SUTD Digital Manufacturing and Design (DManD) Centre,supported by the Singapore National Research Foundation. This research project is partially funded by theSUTD SRG and an internal grant from HKUST (R9429). Furthermore, we would like to thank Sang-InPark from SUTD for help with providing validation results.

19

A. Appendix

A.1. Constitutive coefficients for bilayer laminate cross-sections

We continue the derivation and calculation of the constitutive coefficients for bilayer laminate cross-sections as presented in Sect. 4.1.1.

For the rectangular cross-section with width b and height h, the integrals in (20) can be analyticallyevaluated as:

A033 =

∫ b/2

−b/2

∫ hL

−h/2dx2 dx1 = b

(h

2+ hL

), A1

33 =

∫ b/2

−b/2

∫ h/2

hL

dx2 dx1 = b

(h

2− hL

),

B031 =

∫ b/2

−b/2

∫ hL

−h/2x2 dx2 dx1 =

b

2

(h2L −

h2

4

), B1

31 =

∫ b/2

−b/2

∫ h/2

hL

x2 dx2 dx1 =b

2

(h2

4− h2L

), (A.1)

C011 =

∫ b/2

−b/2

∫ hL

−h/2x22 dx2 dx1 =

b

3

(h3

8+ h3L

), C1

11 =

∫ b/2

−b/2

∫ h/2

hL

x22 dx2 dx1 =b

3

(h3

8− h3L

),

C022 =

∫ b/2

−b/2

∫ hL

−h/2x21 dx2 dx1 =

b3

12

(h

2+ hL

), C1

22 =

∫ b/2

−b/2

∫ h/2

hL

x21 dx2 dx1 =b3

12

(h

2− hL

).

For the circular cross-section with radius r, the analytical evaluation of the integrals in (20) is a littlemore complex. Defining the layer angle θ = 2 arccos(hL/r) and layer width bL = r sin(θ/2) in addition to thelayer height hL, and parameterizing the boundaries of the integrals either as x1(x2) = r sin(arccos(x2/r))or x2(x1) = r cos(arcsin(x1/r)), it follows:

A133 =

∫ bL

−bL

∫ x2(x1)

hL

dx2 dx1 =

∫ bL

−bLr cos(arcsin(x1/r))− hL dx1 =

r2

2(θ − sin θ) ,

A033 = πr2 − A1

33 =r2

2(2π − θ + sin θ) ,

B131 =

∫ bL

−bL

∫ x2(x1)

hL

x2 dx2 dx1 =1

2

∫ bL

−bLr2 cos2(arcsin(x1/r))− h2L dx1 =

2

3r3 sin3(θ/2),

B031 = −B1

31 = −2

3r3 sin3(θ/2), (A.2)

C111 =

∫ bL

−bL

∫ x2(x1)

hL

x22 dx2 dx1 =1

3

∫ bL

−bLr3 cos3(arcsin(x1/r))− h3L dx1 =

r4

16(2θ − sin(2θ)) ,

C011 =

1

4πr4 − C1

11 =r4

16(4π − 2θ + sin(2θ)) ,

C122 =

∫ bL

−bL

∫ x2(x1)

hL

x21 dx2 dx1 =

∫ bL

−bLx21 (r cos(arcsin(x1/r))− hL) dx1 =

r4

48(6θ − 8 sin(θ) + sin(2θ)) ,

C022 =

1

4πr4 − C1

22 =r4

48(12π − 6θ + 8 sin(θ)− sin(2θ)) .

For both cases, rectangular and circular cross-sections, the torsion constant C33 cannot be determinedas a closed-form solution in terms of the parameters hL, b and h resp. r, and E0, E1, and thus have tobe determined through solving a 2-dimensional PDE problem on the cross-section domain S, see [25]. Forcircular cross-sections, we have obtained an approximation from numerical data as C33 = E0 C

033 + E1 C

133

with:

C033 =

1

16πr4

(2(1 + hL/r)

2 + (1 + hL/r)3),

C133 =

1

16πr4

(2(1− hL/r)2 + (1− hL/r)3

),

(A.3)

which was then used in the numerical applications presented.

20

A.2. Constitutive coefficients for continuously graded cross-sections

We continue the derivation and calculation of the constitutive coefficients for continuously graded, rect-angular cross-sections as presented in Sect. 4.1.2. The parameters of the cross-section are width b, heighth and Young’s modulus distribution E(x2) = E0 + (E1 − E0)

(12 + x2

h

)pwith E0, E1, p > 0. Then, the

analytical evaluation of the integrals yields:

A33 =

∫ b/2

−b/2

∫ h/2

−h/2E(x2) dx2 dx1 = bh

pE0 + E1

p+ 1,

B31 =

∫ b/2

−b/2

∫ h/2

−h/2x2E(x2) dx2 dx1 =

bh2

2

p(E1 − E0)

(p+ 1)(p+ 2),

C11 =

∫ b/2

−b/2

∫ h/2

−h/2x22E(x2) dx2 dx1 =

bh3

12

pE0(8 + 3p+ p2) + 3E1(2 + p+ p2)

(p+ 1)(p+ 2)(p+ 3), (A.4)

C22 =

∫ b/2

−b/2

∫ h/2

−h/2x21E(x2) dx2 dx1 =

b3h

12

pE0 + E1

p+ 1,

C33 ≡ C33(b, h,E0, E1, p).

Bibliography

[1] I. Gibson, D. Rosen, and B. Stucker. Additive Manufacturing Technologies: 3D Printing, Rapid Prototyping, and DirectDigital Manufacturing. Springer-Verlag New York, second edition, 2015.

[2] B.G. Compton and J.A. Lewis. 3D-printing of lightweight cellular composites. Advanced Materials, 26(34):5930–5935,2014.

[3] X. Zheng, H. Lee, T.H. Weisgraber, M. Shusteff, J. DeOtte, E.B. Duoss, J.D. Kuntz, M.M. Biener, Q. Ge, J.A. Jackson,S.O. Kucheyev, N.X. Fang, and C.M. Spadaccini. Ultralight, ultrastiff mechanical metamaterials. Science, 344(6190):1373–1377, 2014.

[4] A.P. Garland and G. Fadel. Design and manufacturing functionally gradient material objects with an off the shelf 3Dprinter: Challenges and solutions. ASME Journal of Mechanical Design, 137(11):111407, 2015.

[5] A.D.B.L. Ferrera, P.R.O. Novoa, and A.T. Marques. Multifunctional material systems: A state-of-the-art review. Com-posite Structures, 151:3–35, 2016.

[6] O. Weeger, Y.S.B. Kang, S.-K. Yeung, and M.L. Dunn. Optimal design and manufacture of active rod structures withspatially variable materials. 3D Printing and Additive Manufacturing, 3(4):204–215, 2016.

[7] N. Hu and R. Burgueo. Buckling-induced smart applications: Recent advances and trends. Smart Materials and Structures,24(6):063001, 2015.

[8] Q. Ge, A.H. Sakhaei, H. Lee, C.K. Dunn, N.X. Fang, and M.L. Dunn. Multimaterial 4D printing with tailorable shapememory polymers. Scientific Reports, 6:31110, 2016.

[9] Z. Ding, C. Yuan, X. Peng, T. Wang, H.J. Qi, and M.L. Dunn. Direct 4D printing via active composite materials. ScienceAdvances, 3(4):e1602890, 2017.

[10] Z. Ding, O. Weeger, H.J. Qi, and M.L. Dunn. 4D rods: 3D structures via programmable 1D composite rods. Materials &Design, 137:256–265, 2018.

[11] T.W. Murphey. Large strain composite materials in deployable space structures. In 17th International Conference onComposite Materials, Edinburgh, United Kingdom, 2009. The British Composites Society.

[12] T.W. Murphey, D. Turse, and L. Adams. TRAC boom structural mechanics. In 4th AIAA Spacecraft Structures Confer-ence, Grapevine, TX, United States, 2017. American Institute of Aeronautics and Astronautics.

[13] S. Suresh and A. Mortensen. Fundamentals of functionally graded materials. IOM Communications Limited, 1998.[14] D.H. Hodges. Nonlinear Composite Beam Theory, volume 213 of Progress in Astronautics and Aeronautics. American

Institute of Aeronautics and Astronautics, 2006.[15] B.V. Sankar. An elasticity solution for functionally graded beams. Composites Science and Technology, 61(5):689–696,

2001.[16] Z. Zhong and T. Yu. Analytical solution of a cantilever functionally graded beam. Composites Science and Technology,

67(3–4):481–488, 2007.[17] X.-F. Li. A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–

Bernoulli beams. Journal of Sound and Vibration, 318(4–5):1210–1229, 2008.[18] A. Chakraborty, S. Gopalakrishnan, and J.N. Reddy. A new beam finite element for the analysis of functionally graded

materials. International Journal of Mechanical Sciences, 45:519–539, 2003.[19] S.S. Antman. Nonlinear Problems of Elasticity, volume 107 of Applied Mathematical Sciences. Springer New York, 2005.[20] S. Eugster. Geometric Continuum Mechanics and Induced Beam Theories, volume 75 of Lecture Notes in Applied and

Computational Mechanics. Springer International Publishing, 2015.

21

[21] E. Reissner. On finite deformations of space-curved beams. Zeitschrift fur angewandte Mathematik und Physik ZAMP,32(6):734–744, 1981.

[22] J.C. Simo. A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Computer Methods inApplied Mechanics and Engineering, 49(1):55–70, 1985.

[23] L. Vu-Quoc, H. Deng, and I.K. Ebcioglu. Multilayer beams: A geometrically exact formulation. Journal of NonlinearScience, 6(3):239–270, 1996.

[24] U. Eroglu. Large deflection analysis of planar curved beams made of functionally graded materials using variationaliterational method. Composite Structures, 136:204–216, 2016.

[25] M. Bırsan, H. Altenbach, T. Sadowski, V.A. Eremeyev, and D. Pietras. Deformation analysis of functionally graded beamsby the direct approach. Composites Part B: Engineering, 43(3):1315–1328, 2012.

[26] M. Bırsan, T. Sadowski, L. Marsavina, E. Linul, and D. Pietras. Mechanical behavior of sandwich composite beams madeof foams and functionally graded materials. International Journal of Solids and Structures, 50:519–530, 2013.

[27] L.-L. Ke, J. Yang, and S. Kitipornchai. Nonlinear free vibration of functionally graded carbon nanotube-reinforcedcomposite beams. Composite Structures, 92(3):676–683, 2010.

[28] W. Yu, D.H. Hodges, V. Volovoi, and C.E.S. Cesnik. On Timoshenko-like modeling of initially curved and twistedcomposite beams. International Journal of Solids and Structures, 39(19):5101–5121, 2002.

[29] S. Klinkel and S. Govindjee. Using finite strain 3D-material models in beam and shell elements. Engineering Computations,19(8):902–921, 2002.

[30] S. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill Book Company, Inc., 2nd edition edition, 1951.[31] O.T. Bruhns. Advanced Mechanics of Solids. Springer-Verlag Berlin Heidelberg GmbH, 2003.[32] J. Murın and V. Kutis. 3D-beam element with continuous variation of the cross-sectional area. Computers and Structures,

80:329–338, 2002.[33] J. Murın and V. Kutis. Geometrically non-linear truss element with varying stiffness. Engineering Mechanics, 13(6):435–

452, 2006.[34] V. Kutis, J. Murın, R. Belak, and J. Paulech. Beam element with spatial variation of material properties for multiphysics

analysis of functionally graded materials. Computers and Structures, 89:1192–1205, 2011.[35] G. Balduzzi, M. Aminbaghai, E. Sacco, J. Fussl, J. Eberhardsteiner, and F. Auricchio. Non-prismatic beams: A simple

and effective Timoshenko-like model. International Journal of Solids and Structures, 90:236–250, 2016.[36] K.-J. Bathe and S. Bolourchi. Large displacement analysis of three-dimensional beam structures. International Journal

for Numerical Methods in Engineering, 14(7):961–986, 1979.[37] J.C. Simo and L. Vu-Quoc. A three-dimensional finite-strain rod model. Part II: Computational aspects. Computer

Methods in Applied Mechanics and Engineering, 58(1):79–116, 1986.[38] A. Ibrahimbegovic. On finite element implementation of geometrically nonlinear reissner’s beam theory: three-dimensional

curved beam elements. Computer Methods in Applied Mechanics and Engineering, 122(1):11–26, 1995.[39] M. Bergou, M. Wardetzky, S. Robinson, B. Audoly, and E. Grinspun. Discrete elastic rods. ACM Trans. Graph.,

27(3):63:1–63:12, August 2008.[40] P. Jung, S. Leyendecker, J. Linn, and M. Ortiz. A discrete mechanics approach to the Cosserat rod theory—part 1: static

equilibria. International Journal for Numerical Methods in Engineering, 85(1):31–60, 2011.[41] E. Marino. Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams. Computer Methods

in Applied Mechanics and Engineering, 307:383–410, 2016.[42] O. Weeger, S.-K. Yeung, and M.L. Dunn. Isogeometric collocation methods for Cosserat rods and rod structures. Computer

Methods in Applied Mechanics and Engineering, 316:100–122, 2017.[43] T.J.R. Hughes, J.A. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and

mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194(39–41):4135–4195, 2005.[44] J. A. Cottrell, T. J. R. Hughes, and A. Reali. Studies of refinement and continuity in isogeometric structural analysis.

Computer Methods in Applied Mechanics and Engineering, 196:4160–4183, 2007.[45] R. Bouclier, T. Elguedj, and A. Combescure. Locking free isogeometric formulations of curved thick beams. Computer

Methods in Applied Mechanics and Engineering, 245–246:144–162, 2012.[46] L. Greco and M. Cuomo. B-spline interpolation of Kirchhoff-Love space rods. Computer Methods in Applied Mechanics

and Engineering, 256:251–269, 2013.[47] O. Weeger, U. Wever, and B. Simeon. Isogeometric analysis of nonlinear Euler-Bernoulli beam vibrations. Nonlinear

Dynamics, 72(4):813–835, 2013.[48] F. Auricchio, L. Beirao da Veiga, T.J.R. Hughes, A. Reali, and G. Sangalli. Isogeometric collocation methods. Mathematical

Models and Methods in Applied Sciences, 20(11):2075–2107, 2010.[49] A. Reali and T.J.R. Hughes. An introduction to isogeometric collocation methods. In G. Beer and S. Bordas, editors,

Isogeometric Methods for Numerical Simulation, volume 561 of CISM International Centre for Mechanical Sciences, pages173–204. Springer, 2015.

[50] D. Schillinger, J. A. Evans, A. Reali, M. A. Scott, and T. J. R. Hughes. Isogeometric collocation: Cost comparisonwith Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Computer Methods in AppliedMechanics and Engineering, 267:170–232, 2013.

[51] F. Auricchio, L. Beirao da Veiga, J. Kiendl, C. Lovadina, and A. Reali. Locking-free isogeometric collocation methods forspatial Timoshenko rods. Computer Methods in Applied Mechanics and Engineering, 263:113–126, 2013.

[52] J. Kiendl, F. Auricchio, T.J.R. Hughes, and A. Reali. Single-variable formulations and isogeometric discretizations forshear deformable beams. Computer Methods in Applied Mechanics and Engineering, 284:988–1004, 2015.

[53] F. Auricchio, L. Beirao da Veiga, J. Kiendl, C. Lovadina, and A. Reali. Isogeometric collocation mixed methods for rods.

22

Discrete and Continuous Dynamical Systems - Series S, 9(1):33–42, 2016.[54] G. Balduzzi, S. Morganti, F. Auricchio, and A. Reali. Non-prismatic Timoshenko-like beam model: Numerical solution

via isogeometric collocation. Computers and Mathematics with Applications, 74, 2017.[55] F. Auricchio, L. Beirao da Veiga, T.J.R. Hughes, A. Reali, and G. Sangalli. Isogeometric collocation for elastostatics and

explicit dynamics. Computer Methods in Applied Mechanics and Engineering, 249-252:2–14, 2012. Higher Order FiniteElement and Isogeometric Methods.

[56] J. Kiendl, F. Auricchio, L. Beir ao da Veiga, C. Lovadina, and A. Reali. Isogeometric collocation methods for the Reissner-Mindlin plate problem. Computer Methods in Applied Mechanics and Engineering, 284:489–507, 2015. IsogeometricAnalysis Special Issue.

[57] F. Maurin, F. Greco, L. Coox, D. Vandepitte, and W. Desmet. Isogeometric collocation for Kirchhoff–Love plates andshells. Computer Methods in Applied Mechanics and Engineering, 329:396–420, 2018.

[58] R. Kruse, N. Nguyen-Thanh, L. De Lorenzis, and T.J.R. Hughes. Isogeometric collocation for large deformation elasticityand frictional contact problems. Computer Methods in Applied Mechanics and Engineering, 296:73–112, 2015.

[59] J. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs. Isogeometric Analysis: Toward Integration of CAD and FEA. JohnWiley & Sons, Ltd, 2009.

[60] L.A. Piegl and W. Tiller. The NURBS Book. Monographs in Visual Communication. Springer, 1997.[61] E. Marino. Locking-free isogeometric collocation formulation for three-dimensional geometrically exact shear-deformable

beams with arbitrary initial curvature. Computer Methods in Applied Mechanics and Engineering, 324:546–572, 2017.[62] T.A. Davis. Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Transactions on

Mathematical Software (TOMS), 30(2):196–199, 2004.[63] O. Weeger, N. Boddeti, S.-K. Yeung, S. Kaijima, and M.L. Dunn. Digital design and nonlinear simulation for additive

manufacturing of soft lattice structures. Additive Manufacturing, 2018. Under review.

23